Example 10.28.5. Let us show that the family of principal ideals of a ring R is an Oka family. Indeed, suppose I \subset R is an ideal, a \in R, and (I, a) and (I : a) are principal. Note that (I : a) = (I : (I, a)). Setting J = (I, a), we find that J is principal and (I : J) is too. By Lemma 10.28.1 we have I = J (I : J). Thus we find in our situation that since J = (I, a) and (I : J) are principal, I is principal.
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